Let $(Q_n)$ be probability measures on some metric space $(S,\mathcal{B}_S)$ and $\mathcal{H}\subset\mathcal{B}_S$ a separating class. $\partial A$ denotes the boundary of $A$. If I have:
(1) $\lim_{n\to\infty}Q_n[A]=Q[A]$ for all $A\in\mathcal{H}$ with $Q[\partial A]=0$
(2) $\lim_{n\to\infty}Q_n[A]=\mu[A]$ for all $A\in\mathcal{H}$ with $\mu[\partial A]=0$
Can I conclude that $Q=\mu$?